Japanese Monetary Policy Reaction Function and Time-Varying … · 2017. 3. 6. · Japanese...

Japanese Monetary Policy Reaction Function andTime-Varying Structural Vector Autoregressions:

A Monte Carlo Particle Filtering Approach ∗

Koiti Yano† Naoyuki Yoshino‡

Abstract

Recent years, the Japanese monetary policy is one of the hot topics on the Japan economy. This paper

presents the empirical analysis on Japanese monetary policy based on Time-Varying Structural Vector AutoRegres-

sions (TVSVAR). Our TVSVAR includes a monetary reaction function, an aggregate supply function, an aggregate

demand function, and real exchange rate determination function. Our TVSVAR is a dynamic full recursive struc-

tural VAR, which is similar to Primiceri (2005), Canova and Gambetti (2006), and many related papers. The most

of previous studies on TVSVAR are based on Markov Chain Monte Carlo method and the Kalman filter. We, how-

ever, adopt a new TVSVAR estimation method that is based on the Monte Carlo Particle filter and a self-organizing

state space model, proposed by Kitagawa (1996), Gordon et al. (1993), Kitagawa (1998), Yano (2007b), and Yano

(2007a). The method is proposed by Yano (2007c). Our methods are applied for the estimation of a quarterly

model of the Japanese economy (a nominal short term interest rate, the rate of inflation, the growth rate of real

output, and the real effective exchange rate). We would like to emphasize that our paper is the first one to analyze

the Japanese economy using TVSVAR. It is often asked the causes of long term recession of the Japanese economy

in 1990s whether it is caused by aggregate supply factor or aggregate demand factor. This paper concludes that

both supply and demand factors contribute to the 10-years recession.

Key words : Monte Carlo particle filter, Self-organizing state space model, time-varying coefficient,

structural vector autoregressions, Markov chain priors.

∗The authors would like to thank Gianni Amisano, Yuzo Honda, Cheng Hsiao, Yasuyuki Iida, Genshiro Kitagawa, Naoto

Kunitomo, Munehisa Kasuya, Kiyohiko Nishimuara, Jack Li, Warwick McKibbin, Ryuzo Miyao, Anwar Nasution, Yasushi

Okada, Yasuhiro Omori, Wolfgang Polasek, Ayano Sato, Seisho Sato, Katsuhiro Sugita, Yoshiyasu Tamura, Akira Terai, Ha-

jime Wago, Helga Wagner, Toshiaki Watanabe, Sebastian Watzka, participants of 2nd Japanese-European Bayesian Econo-

metrics and Statistics Meeting, Japan Society of Monetary Economics Autumn Meeting 2007, Hitotsubashi Conference on

Econometrics 2007, and Asian Economic Panel Meeting 2007 for their helpful comments. The authors would like to thank

the Institute of Statistical Mathematics for the facilities and the use of SGI Altix3700 and HP XC4000. This paper presents

the authors’ personal views, which are not necessarily the official ones of the Financial Research and Training Center or the

Financial Services Agency.†Financial Services Agency, Government of Japan. E-mail: [email protected], [email protected]‡Faculty of Economics, Keio University

1 Introduction

Recent years, Japanese monetary policy is one of the hot topics on the Japanese economy. This paper presents

the empirical analysis on Japanese monetary policy using Time-Varying Structural Vector AutoRegressions (TVS-

VAR). Our TVSVAR includes a monetary reaction function, an aggregate supply function, an aggregate demand

function, and real exchange rate determination function. The changes of coefficients indicate the changes of the

correlations of macroeconomic variables. Thus, we are able to analyze the changes of the Japanese economy.

Our approach is related to Uhlig (1997), Cogley and Sargent (2001), Ciccarelli and Rebucci (2003), Cogley and

Sargent (2005), Primiceri (2005), Sims and Zha (2006), Canova and Gambetti (2006), and many studies. The most

of previous studies are based on Markov Chain Monte Carlo method and the Kalman filter4. In the studies, the

random walk priors (the Minnesota priors), which are based on linear Gaussian state space modeling, are assumed

on the time-evolutions of coefficients. The priors are proposed by Doan et al. (1984)5. Yano (2007c), however,

proposes a new TVSVAR estimation method that is based on the Monte Carlo Particle filter and a self-organizing

state space model, proposed by Kitagawa (1996), Gordon et al. (1993), Kitagawa (1998), Yano (2007b), and Yano

(2007a). A novel feature of Yano (2007c) is that it assumes the time evolutions of coefficients are given by Markov

chain processes. We call this assumptionMarkov chain priorson time-varying coefficients. Our priors are based

on the nonlinear non-Gaussian state space modeling. The linear Gaussian cases of the Markov chain priors are

equivalent to the random walk priors. Thus, our method is more flexible rather than previous methods. Our method

is applied for the estimation of a quarterly model of the Japanese economy (a nominal short term interest rate, in-

flation rate, real growth rate, and the return of the real effective exchange rate). We detect structural changes in

most coefficients of TVSVAR.

There exist previous studies on the Japanese monetary policy based on Bayesian statistical approach: Kimura

et al. (2003), Fujiwara (2006), and Inoue and Okimoto (2007)6. Kimura et al. (2003) estimates time-varying

reduced-form VAR models based on the Kalman filter. Fujiwara (2006) and Inoue and Okimoto (2007) analyze the

regime changes of the Japanese economy in 1990s using Markov Switching VAR (MSVAR). The main advantages

of our method to the previous studies are the we need less restrictions on the time-evolution of coefficients and less

prior knowledge on structural changes. Kimura et al. (2003) assume random walk priors (the Minnesota priors) on

the time-evolution of coefficients, which are based on linear Gaussian state space modeling. We, however, adopt

Markov chain priors, which assume the the time-evolutions of coefficients follow Markov chain processes. Our

assumption is less restricted rather than random walk priors. Fujiwara (2006) and Inoue and Okimoto (2007),

use prior knowledge on the number of structural changes of the Japanese economy. In our method, the structural

changes of coefficients of the economy are detected using the estimated time-varying coefficients of our model.

Thus, we don’t need prior knowledge on the structural changes of coefficients and the regime changes of the

4Canova (2007) and Dejong and Dave (2007) are introductory textbooks on Bayesian statistical approach for macroeco-

nomic analysis. Fernandez-Villaverde and Rubio-Ramirez (2005) and Fernandez-Villaverde and Rubio-Ramirez (2007) have

shown that the Monte Carlo particle filter and Metropolis-Hastings algorithm can be successfully applied to estimate DSGE

models.5The random walk priors are equivalent to first-order smoothness priors, proposed by Kitagawa (1983). TVSVAR based on

the Kalman filter is adopted in Jiang and Kitagawa (1993) and Yano (2004) to estimate reduced-form time-varying coefficients

vector autoregressions.6Miyao (2006) is a comprehensive survey on the Japanese macroeconomic and monetary policy based on structural VARs.

Kasuya and Tanemura (2000) constructs Bayesian VAR optimized by the Posterior Information Criterion and estimates the

performance of forecasting.

1

Japanese economy. Moreover, an advantage of our approach to the MSVAR approach is that it is not necessary for

us to divide our date set into multi-pieces, even if several structural changes happen in the data. In the MSVAR

approach, for example, if a structural change happens in your data set, you need to divide the data into two pieces

to estimate each VAR for each piece. In general, the size of macroeconomic data is relatively small. Thus, this

problem may cause poor estimation if several structural changes happen in your data set. In our approach, however,

this problem doesn’t happen. You are able to use your whole data set to estimate TVSVAR, even if several structural

changes happen in it.

Major findings of this paper are summarized as follows. (i) Monetary policy by changing the interest rate

worked well to control real GDP in 1980s. However, it did not work to control real GDP since 1990s. Furthermore

lower interest rate brought lower economic growth. (ii) The rate of interest show almost no impact on rate of

inflation after 1990 even though interest rate policy worked to control inflation in 1980s. (iii) Policy reaction

of the interest rate to rate of inflation was strong in early 1980s. However the interest rate reaction to rate of

inflation diminished drastically after 1997. Especially introduction of zero interest rate policy bounded by zero

and the central bank of Japan could not set its interest rate into negative value. (iv) It is often asked the causes of

long term recession of the Japanese economy in 1990s whether it is caused by aggregate supply factor (Hayashi

and Prescott (2002), Hayashi (2003) and Miyao (2006)) or aggregate demand factor (such papers as Kuttner and

Posen (2001) and Kuttner and Posen (2002)). This paper will show both supply and demand factors contributed

to the 10-years recession. (v) From estimate of aggregate demand, there can be found spiral effects in Japanese

economy especially after 1995. Lower real GDP and lower rate of inflation accelerated sluggish economy and

created downward spiral. (vi) From estimate of aggregate supply, the spiral effects are not observed in a sense that

lower inflation did not cause much further decline in rate of inflation.

This paper is organized as follows. In section 2, we describe Time-Varying Structural Autoregressions and

the outline of a new TVSVAR estimation method, proposed by Yano (2007c). In section 3, we show empirical

analyses on the Japanese monetary policy and the Japanese economy. In section 4, we describe conclusions and

discussions.

2 Time-Varying Structural Vector Autoregressions

In this section, we describe the outline of Yano (2007c). First, we describe Time-Varying Structural Vector Au-

toregressions and define state vectors to estimate it. Second, we explain the Monte Carlo particle filter and a

self-organizing state space model to estimate a nonlinear non-Gaussian state space model.

2.1 Time-Varying Structural Vector Autoregressions

Time-Varying Structural Vector Autoregressions (TVSVAR) for the time seriesY1:T = {Y1,Y2, · · · , YT } is de-

fined as follows.

B0,tYt =P∑

p=1

Bp,tYt−p + Dtut−κ + ct + ϵt, (1)

whereYt is a(k × 1) vector of observations at timet, ut−κ is an(n× 1) vector of disturbances at timet, κ ≥ 1 is

a constant,ct is a(k × 1) vector of time-varying intercepts at timet, andϵt =(ϵ1,t, · · · , ϵk,t

)T ∼ N(0, V ) with

2

V = diag(σ21 , σ2

2 , · · · , σ2k) 7 . The matrices of time varying coefficients are

B0,t =

1 0 · · · 0

−b2,1,0,t 1 · · · 0...

. ... ..

...

−bk,1,0,t · · · −bk,k−1,0,t 1

, (2)

Bp,t =

b1,1,p,t · · · b1,k,p,t

.... . .

...

bk,1,p,t · · · bk,k,p,t

, (3)

and

Dt =

d1,1,t · · · d1,n,t

.... ..

...

dk,1,t · · · dk,n,t

. (4)

Jiang and Kitagawa (1993) pointed out that Eq. (1) can be estimated by the each component ofYt becauseV is a

diagonal matrix. For example,Y1,t,the first component ofYt in Eq. (1), can be written by

Y1,t =b1,1,1,tY1,t−1 + b1,2,1,tY2,t−1 · · · + b1,k,1,tYk,t−1+

· · · + b1,1,p,tY1,t−1 + b1,2,p,tY2,t−1 · · · + b1,k,p,tYk,t−1 + c1,t + ϵ1,t,(5)

wherec1,t is the first component ofct, andϵ1,t is the first component ofϵt. For another example,Y2,t, the second

component ofYt in Eq. (1), can be written by

Y2,t =b2,1,0,tY1,t + b2,1,1,tY1,t−1 + b2,2,1,tY2,t−1 · · · + b2,k,1,tYk,t−1+

· · · + b2,1,p,tY1,t−1 + b2,2,p,tY2,t−1 · · · + b2,k,p,tYk,t−1 + c2,t + ϵ2,t,(6)

wherec2,t is the second component ofct, andϵ2,t is the second component ofϵt. For the first example, we define

a state vectorxt of time varying coefficients as follows.

xt =[b1,1,1,t, b1,2,1,t, · · · , b1,k,1,t, · · · , b1,1,p,t, b1,2,p,t, · · · b1,k,p,t, c1,t

]T

. (7)

For the second example, we define another state vectorxt of time varying coefficients as follows.

xt =[b2,1,0,t, b2,1,1,t, b2,2,1,t, · · · , b2,k,1,t, · · · , b2,1,p,t, b2,2,p,t, · · · b2,k,p,t, c2,t

]T

. (8)

According to the discussion which is described above, the main problem of TVSVAR is how to estimate the state

vectorxt. In the framework of sequential Bayesian filtering, the filtering distribution ofxt, which is based on the

observations,Y1:t, is given by

p(xt|Y1:t). (9)

The smoothing distribution ofxt, which is based on the observations,Y1:T , is given by

p(xt|Y1:T ). (10)

7In this paper, a bold-faced symbol means a vector or a matrix.

3

Moreover, we assume that the time evolution ofxt is given by

p(xt|xt−1). (11)

We refer to this assumption asMarkov Chain priorson time-varying coefficients. Our priors are based on the

nonlinear non-Gaussian state space modeling. The linear Gaussian cases of the Markov chain priors are equivalent

to random walk priors, which are often adopted in previous studies. We would like to emphasize our Markov chain

priors overcome the restriction of random walk priors. Our problem is how to estimate the state vectorxt using

Eq. (9), (10), and (11). To solve the problem, we adopt the Monte Carlo particle filter. In the next subsection, we

describe a method to estimate the state vectorxt using the filter.

2.2 Nonlinear Non-Gaussian State Space Modeling and A Self-Organizing State SpaceModel

To estimate a state vectorxt, we adopt the Monte Carlo Particle Filter (MCPF), proposed by Kitagawa (1996) and

Gordon et al. (1993) and a self-organizing state space model, proposed by Kitagawa (1998). In this subsection, we

describe a nonlinear non-Gaussian state space model and a self-organizing state space model (MCPF is described

in the next subsection).

A nonlinear non-Gaussian state space model for the time seriesYt, t = {1, 2, · · · , T} is defined as follows.

xt = f(xt−1) + vt,

Yt = ht(xt) + ϵt,(12)

wherext is an unknownnx × 1 state vector,vt is nv × 1 system noise vector with a density functionq(v|·) 8, ϵt is

nϵ × 1 observation noise vector with a density functionr(ϵ|·). The functionf : Rnx ×Rnv → Rnx is a possibly

nonlinear function and the functionht : Rnx × Rnϵ → Rny is a possibly nonlinear time-varying function. The

first equation of (12) is called a system equation and the second equation of (12) is called an observation equation.

A system equation depends on a possibly unknownns × 1 parameter vector,ξs, and an observation equation

depends on a possibly unknownno × 1 parameter vector,ξo. This nonlinear non-Gaussian state space model

specifies the two following conditional density functions.

p(xt|xt−1, ξs),

p(Yt|xt, ξo).(13)

Note thatp(xt|xt−1, ξs) is equivalent to Eq. (11). We define a parameter vectorθ as follows.

θ =

ξs

ξo

. (14)

We denote thatθj is thejth element ofθ andJ(= ns +no) is the number of elements ofθ. This type of state space

model (12) contains a broad class of linear, nonlinear, Gaussian, or non-Gaussian time series models. In state space

modeling, estimating the state space vectorxt is the most important problem. For the linear Gaussian state space

model, the Kalman filter, which is proposed by Kalman (1960), is the most popular algorithm to estimate the state

vectorxt. For nonlinear or non-Gaussian state space model, there are many algorithms. For example, the extended

Kalman filter (Jazwinski (1970)) is the most popular algorithm and the other examples are the Gaussian-sum filter8The system noise vector is independent of past states and current states.

4

(Alspach and Sorenson (1972)), the dynamic generalized model (West et al. (1985)), and the non-Gaussian filter

and smoother (Kitagawa (1987)). In recent year, MCPF for nonlinear non-Gaussian state space model is a popular

algorithm because it is easily applicable to various time series models9.

In econometric analysis, generally, we don’t know the parameter vectorθ. In the framework of TVSVAR,

the unknown parameter vectors areξo andξs. In traditional parameter estimation, maximizing the log-likelihood

function ofθ is often used. The log-likelihood ofθ in MCPF is proposed by Kitagawa (1996). However, MCPF is

problematic to estimate the parameter vectorθ because the likelihood of the filter contains error from the Monte

Carlo method. Thus, you cannot use nonlinear optimizing algorithm like Newton’s method10. To solve the

problem, Kitagawa (1998) proposes a self-organizing state space model. In Kitagawa (1998), an augmented state

vector is defined as follows.

zt =

xt

θ

. (15)

An augmented system equation and an augmented measurement equation are defined as

zt = F (zt−1, vt, ξs),

Yt = Ht(zt, ϵt, ξo),(16)

where

F (zt−1, vt, ξs) =

f(xt−1) + vt,

θ

and

Ht(zt, ϵt, ξo) = ht(xt) + ϵt.

This nonlinear non-Gaussian state space model is called a self-organizing state space (SOSS) model. This self-

organizing state space model specifies the two following conditional density functions.

p(zt|zt−1),

p(Yt|zt).(17)

2.3 The Monte Carlo Particle Filter

Most algorithms of sequential Bayesian filtering are based on Bayes’ theorem (See Arulampalam et al. (2002)),

which is

P(zt|Y1:t) =P(Yt|zt)P(zt|Y1:(t−1))

P(Yt|Y1:(t−1)), t ≥ 1, (18)

whereP(zt|Y1:t−1) is the prior probability,P(Yt|zt) is the likelihood,P(zt|Y1:t) is the posterior probability, and

P(Yt|Y1:t−1) is the normalizing constant. We denote an initial probabilityP(z0) = P(z0|∅), where the empty

set∅ indicates that we have no observations. In the state estimation problem, determining an initial probability

P(z0), which is called filter initialization, is important because a proper initial probability improves a posterior

probability. In TVSVAR, an initial probability is restricted in−1 < xi,0 < 1 , wherexi,0 is theith element ofx0.

In MCPF, the posterior density distribution at timet is approximated as

p(zt|Y1:t) =1∑M

m=1 wmt

M∑m=1

wmt δ(zt − zm

t ), (19)

9Many applications are shown in Doucet et al. (2001).10See Yano (2007b).

5

wherewmt is the weight of a particlezm

t , M is the number of particles, andδ is the Dirac’s delta function11. The

definition ofwmt is described below. In the standard algorithm of MCPF, particles are resampled with sampling

probabilities proportional to the weightswmt at every timet. It is necessary to prevent increasing the variance of

weights after few iterations of Eq. (18)12. After resampling, the weights are reset towmt = 1/M . Therefore, Eq.

(19) is rewritten as

p(zt|Y1:t) =1M

M∑m=1

δ(zt − zmt ) (20)

wherezmt are particles after resampling. Using Eq. (20), the predictorp(zt|Y1:(t−1)) can be approximated by

p(zt|Y1:(t−1)) =∫

p(zt|zt−1)p(zt−1|Y1:(t−1))dzt−1

=1M

M∑m=1

∫p(zt|zt−1)δ(zt−1 − zm

t−1)dzt−1

=1M

M∑m=1

p(zt|zmt−1)

≃ 1M

M∑m=1

δ(zt − zmt ).

(21)

Note thatzmt are obtained from

zmt ∼ p(zt|zm

t−1). (22)

Substituting Eq. (21) to Eq. (18), we obtain the following equation.

p(zt|Y1:t) ∝p(Yt|zt)p(zt|Y1:(t−1))

∝ 1M

p(Yt|zt)M∑

m=1

δ(zt − zmt )

=1M

M∑m=1

p(Yt|zmt )δ(zt − zm

t ).

(23)

Comparing Eq. (19) and Eq. (23) indicates that weightswmt are obtain by

wmt ∝ p(Yt|zm

t ). (24)

Therefore, a weightwmt is defined as

wmt ∝ p(Yt|zm

t ) = r(ψt(Yt, zmt ))

∣∣∣∂ψ

∂y

∣∣∣,m = {1, · · · ,M}, (25)

whereψt is the inverse function of the functionht13. In our TVSVAR estimation method, the augmented state

vector is estimated using MCPF. Thus, states and parameters are estimated simultaneously without maximizing

the log-likelihood of Eq. (16) because the parameter vectorθ in Eq. (16) is approximated by particles and it is

11The Dirac delta function is defined as

δ(x) = 0, if x = 0,Z ∞

∞δ(x)dx = 1.

12See Doucet et al. (2000).13See Kitagawa (1996).

6

estimated as the state vector in Eq. (15)14. The algorithm of our TVSVAR estimation method is summarized as

follows 15.Algorithm: Time-Varying Structural Vector Autoregressions Estimation

SOSS[{zmt−1}

M

m=1, yt]

{FOR m=1,...M

Predict:zmt ∼ p(zt|zm

t−1, vmt )

Weight:wmt is obtained by Eq. (25)

ENDFOR

Sum of Weights:sw =∑M

m=1 wit

Log-Likelihood: llk = log(sw/M)

FOR m=1,...,M

Normalize:wmt = wm

t

sw

ENDFOR

Resampling: [{zmt , wm

t }Mm=1] =resample[{zm

t , wmt }M

m=1]

RETURN[{zmt , wm

t }Mm=1, llk]

}SOSS.MAIN[{xm

0 }Mm=1, {y}T

t=1, P ]

{θ0 ∼ uniform(P − r,P + r)

{zm0 }M

m=1 = ({xm0 }M

m=1, {θm0 }M

m=1)

FOR t=1,...,T

soss = SOSS[{zmt−1}

M

m=1, yt]

{zmt , wm

t }Mm=1 = ({zm

t , wmt }M

m=1 in soss)

ENDFOR

RETURN[{{zmt , wm

t }Mm=1}

T

t=1, Y ]

}

On a self-organizing state space model, however, Hurseler and Kunsch (2001) points out a problem: determi-

nation of initial distributions of parameters for a self-organizing state space model. The estimated parameters of a

self-organizing state space model comprise a subset of the initial distributions of parameters. We must know the

posterior distributions of parameters to estimate parameters adequately. However, the posterior distributions of the

parameters are generally unknown. Parameter estimation fails if we do not know appropriate their initial distribu-

tions. Yano (2007b) proposes a method to seek initial distributions of parameters for a self-organizing state space

model using the simplex Nelder-Mead algorithm to solve the problem. To seek initial distributions of parameters,

we adopt the algorithm, which is proposed by Yano (2007b). Moreover, we adopt the smoothing algorithm and

filter initialization method, which is proposed by Yano (2007a).

14The justification of an SOSS model is described in Kitagawa (1998).15The details of MCPF and SOSS is described in Yano (2007b).

7

2.3.1 Functional Forms

In this paper, we use linear non-Gaussian state space models to estimate time-varying coefficients and param-

eters. A linear non-Gaussian state is given by

xt = xt−1 + vt,

Yi,t = Htxt + ϵi,t,(26)

whereYi,t is an observation,vt ∼ q(vt|ξt), ϵi,t ∼ ri(ϵi,t|ξi,o), ϵi,t is theith component ofϵt, andξi,o is theith

component ofξo. The details ofxt andHt are described in Appendix C. In our Markov chain priors,q(vt|ξt)

andri(ϵi,t|ξi,o) are possibly non-Gaussian distributions. We would like to emphasize that our priors make the

estimation of TVSVAR flexible rather than random walk priors. In this paper, the innovation termq(vt|ξt) is

specified by normal distributions ort-distributions, andri(ϵi,t|ξi,o) is specified by normal distributions. In general,

the components,{ξ1,s, ξ2,s, · · · , ξL,s}, of ξs are different (L is defined in appendix C). In this paper, however, to

reduce computational complexity, we assume as follows.

(A1) ξ1,s = ξ2,s = · · · = ξL,s = |ξs| (27)

(A2) ξs = σs (28)

In this paper, the time evolutions of coefficients are given by

xi,t = xi,t−1 + |ξs| × t(df), (29)

wheredf is the degree of freedom of Student’st-distribution. The innovation term ofYi is given by the normal

distributions (ϵi,t ∼ N(0, σ2s)) 16.

3 Empirical Analyses

Our methods are applied for the estimation of a quarterly model of the Japanese economy. In the model, four

variables are included : a short-term interest rate (the uncollateralized overnight call rate), the rate of inflation (the

growth rate of seasonal adjusted GDP deflator), the growth rate of output (the growth rate of seasonal adjusted real

GDP), and the return of the real effective exchange rate17. We use data from 1980:I up to 2006:III. Rate hikes are

given by the first difference of the mean of the monthly average of the uncollateralized overnight call rate18. The

growth rates of GDP deflator and real output are given by

xt =[log Xt − log Xt−1

]× 100. (31)

The growth rate of the real effective exchange rate is given by

et = −[log Et − log Et−1

]× 100, (32)

whereEt is the real effective exchange rate. Note thatet becomes smaller when Yen is appreciated.

In Fig. 1, the four variables are shown.

[Figure 1 about here.]

16Primiceri (2005) proposes TVSVAR with Stochastic Volatility. An elements of Time-Varying Variance Covariance Matrix

is given by

(B2) σi,m,t = |σi,m,t−1 + ηt|, (30)

whereηt ∼ N(0, ξ2i,o).

17The details of data are described in Appendix A.18See Miyao (2000), Miyao (2002), and Miyao (2006).

8

3.1 Full Recursive TVSVAR

We estimate the first order of full recursive TVSVAR (FR-TVSVAR(1)) as a benchmark model. FR-TVSVAR(1)

is given by

it = b1,1,1,tit−1 + b1,2,1,tπt−1 + b1,3,1,tyt−1 + b1,4,1,tet−1 + c1,t + ϵi,t, (33)

πt = b2,1,0,tit + b2,1,1,tit−1 + b2,2,1,tπt−1 + b2,3,1,tyt−1 + b2,4,1,tet−1 + c2,t + ϵπ,t, (34)

yt = b3,1,0,tit + b3,2,0,tπt + b3,1,1,tit−1 + b3,2,1,tπt−1 + b3,3,1,tyt−1 + b3,4,1,tet−1 + c3,t + ϵy,t, (35)

et = b4,1,0,tit + b4,2,0,tπt + b4,3,0,tyt + b4,1,1,tit−1 + b4,2,1,tπt−1 + b4,3,1,tyt−1 + b4,4,1,tet−1 + c4,t + ϵe,t.

(36)

whereit is the first difference of the short-term interest rate,πt is the rate of inflation,yt is the growth rate of

real output, andet is the return of the real effective exchange rate19. Following Miyao (2000), Miyao (2002), and

Miyao (2006), the variables of FR-TVSVAR(1) are ordered from exogenous variables to endogenous variables20.

First, we estimate TVSVAR based on quarterly data of the Japanese economy from 1980:Q1 to 1998:Q4 to

avoid the periods of zero-interest rate policy and quantitative easing policy. Fig. 2, 3, 4, and 5 show Eq. (33),

(34), (35), and (36), respectively. In these figures, the solid line is a estimate of a time-varying coefficient and the

dash-lines are68% confidence interval21.





In Fig. 6, 7, 8, and 9, we show Impulse Response Functions (1980:Q4、1985:Q4、1989:Q4、1997:Q4), respectively.





19We set the number of particles,M , to 10000. Moreover, we set the degrees of freedom,df , in Eq. (29) to 10, 20, 30. In

the all cases, we get same results. In our paper, we show results that we setdf to 10. The other parameters of simulation are

same in Yano (2007b) and Yano (2007a). All time-varying coefficients are standardized as follows:

bx,y,z,t = sdexp/sdobs,

wheresdexp is the standard deviation of an explaining variable andsdexp is the standard deviation of an observation (this

standardization method may not be best).20The results of block recursive TVSVAR, which is a dynamic version of SVAR, proposed by Christiano et al. (1999), are

shown.21Confidence interval is calculated using100 times estimation of a time-varying coefficient.

9

In Fig 10 and 11, we show Quantile-Quantile plot and autocorrelation of residuals of FR-TVSVR(1), respectively.



Second, we estimate TVSVAR based on quarterly data of the Japanese econmy from 1980:Q1 to 2006:Q3. Fig.

12, 13, 14, and 15 show Eq. (33), (34), (35), and (36), respectively.





In Fig 16 and 17, we show Quantile-Quantile plot and autocorrelation of residuals of FR-TVSVR(1), respectively.



We compare TVSVAR with (invariant coefficient) Structural VAR (SVAR) using residual analysis. SVAR(P)

is given by

B0Yt = B1Yt−1 + B2Yt−2 + · · · + BP Yt−P + ϵt

First, we estimate the first order of SVAR (SVAR(1)) using quarterly data of the Japanese economy from 1980:Q1

to 1998:Q4 to avoid the periods of zero-interest rate policy and quantitative easing policy.B0 of SVAR(1) is

B0 =

1 0 0 0

0.0648 1 0 0

−0.5243 0.0319 1 0

−0.6978 0.1627 −0.8602 1

. (37)

The standard error ofB0 is

BSE0 =

1 0 0 0

0.2515 1 0 0

0.2519 0.2241 1 0

0.2604 0.2242 0.1259 1

. (38)

Eq. (38) shows the most of the standard errors inB0 are larger than the elements ofB0. In Table 1, we show the

Root Mean Square Error of FR-TVSVAR(1) and SVAR(1).

[Table 1 about here.]

In Fig 18 and 19, we show Quantile-Quantile plot and autocorrelation of residuals of the first order of Full Recur-

sive Structural Vector Autoregressions, respectively.

10



Second, we estimate SVAR(1) using quarterly data of the Japanese economy from 1980:Q1 to 2006:Q3.B0

of SVAR(1) is

B0 =

1 0 0 0

0.0490 1 0 0

−0.4947 0.0100 1 0

−0.7272 −0.165 −0.7892 1

. (39)

The standard error ofB0 is

BSE0 =

1 0 0 0

0.2481 1 0 0

0.2483 0.1910 1 0

0.2548 0.1910 0.1157 1

. (40)

In Table 2, we show the Root Mean Square Error of FR-TVSVAR(1) and SVAR(1).

[Table 2 about here.]

In Fig 20 and 21, we show Quantile-Quantile plot and autocorrelation of residuals of the first order of Full Recur-

sive Structural Vector Autoregressions, respectively.



In Fig. 2, the results of Eq. (33) are shown. Equation (32) represents the monetary policy reaction of the

Central Bank of Japan. The call lending rate which is controlled by the Central Bank of Japan assumed to depend

on the following four variables, namely, (i) lagged interest rate which represents the smooth adjustment of the

interest rate, (ii) rate of inflation, (iii) the growth rate of real GDP (yt−1) and (iv) the exchange rate. Target values

of the rate of inflation, log of GDP and the exchange rate are captured in the changes in the constant termc1,t.

Page 20 figures (1) to (5) show the changes in the value of coefficients of Equation (32). Figure2 (1) shows that

the central bank of Japan was conducting gradual adjustment of the short term interest rate during 1980 to 1983

when the high rate of inflation in the second oil crisis was observed. However, the Central Bank of Japan stopped

to conduct gradual interest rate policy adjustment during the bubble period (1984 to 1990). The Central bank again

came back to do gradual interest rate adjustment after 1993 to the recent where low (or zero) interest rate policy

was adopted. During the bubble period, the Central bank’s call rate control can be seen as abnormal compared

with other period. Page 20 Figure (2) shows changes in the coefficient of the lagged rate of inflation (πt−1). It

was positive and quite significant between 1980 to 1982 when the high rate of inflation hit Japan. It denotes that

the central bank was watching the rate of inflation as the major target of the monetary policy. However, it turns

negative value during the bubble period which suggests the Central Bank of Japan lowered its call lending rate

despite low rate of inflation during the period of asset price bubble. Page 20, Figure (3) shows the monetary policy

11

reaction to real GDP. When Japanese economy faced with sluggish growth, the call lending rate was raised. Since

the rate of inflation was the main target of the Central Bank monetary policy in early 1980s. Page 20, Figure (4)

denotes the monetary policy reaction to the lagged exchange rate. During the period of early 1980s, despite the

yen appreciation the call lending rate was raised in order to fight against higher rate of inflation. Page 20, Figure

(5) is the fluctuation of the constant term. In our paper, it suggests the level of the call lending rate. In the period of

early 1980s, the level of the call lending rate was high. On the other hand, it had been lowered during the bubble

period and early 1990s. The response of the interest rate by the Central Bank of Japan shows no major reactions

to any indicators since late 1990s until recent. In Fig. 4, the results of Eq. (35) are shown. Figure 3, (1) is the

reaction of the rate of inflation to current nominal interest rate. Higher rate of Inflation raised current nominal

interest rate in 198s and early 1990s. Figure 3, (2) is the response to the lagged interest rate of rate of inflation. In

early　 1980s,　 tight　 monetary　 policy　 led　 to　 lower　 rate　 of　 inflation　 which　 is　

described　 as　 negative　 coefficient of lagged interest rate. Figure 3, (3) is the response to the lagged rate of

inflation. During the　 asset　 bubble　 period　 and　 early　 1990s　 show　 negative　 coefficients.　

It 　 suggests　 despite　 positive　 expected rate of inflation, actual rate of inflation was declining. Figure 3,

(4) is the response of lagged real GDP (yt−1) to the rate of inflation. In mid 1980s, relatively higher growth of the

economy brought higher rate of inflation. On the other hand, influence of real GDP to the rate of inflation was quite

small. Figure 3, (5) is the response of lagged exchange rate (et−1) to the rate of inflation. In early 1980s the rate

of inflation went on going despite higher appreciation of the yen due to the effect of the second oil crisis. In mid

1980s, high appreciation of the yen brought lower rate of inflation which was one of the causes of the asset price

bubble in Japan. Figure 4 shows the aggregate demand function. Figure 4 (3) shows the response of the lagged

interest rate on aggregate demand. In 1986, the lower interest rate pushed economic growth. However, in the

1990, lower interest rate kept sluggish economy which is denoted by the positive sign of the lagged interest rate on

real GDP. Figure 4 (4) is the response of lagged inflation on aggregate demand. In 1983-85 period, real economy

was growing. However, in 1990s, rate of inflation turns to negative figures despite positive low growth rate. Most

recent period shows strong positive sign since the rate of inflation and the economic growth turned positive each

other. Figure 4 (5) is the response of lagged real GDP on aggregate demand. Figure 4 (6) is the response of lagged

exchange rate on aggregate demand. In 1986, despite the appreciation of the yen, real GDP was rising. In 1994

high appreciation of the yen brought slower growth. Figure 5 shows that during the bubble period, inflation had

been lowered and at the same time, inflation was stable. Higher economic growth brought appreciation of the yen.

Equation (33) and Figure 3 show the aggregate supply function and their changes in the coefficients. Since

it is assumed Cholesky decomposition, the current interest rate (it) appears in Equation (33). The coefficient of

it is positive which would be showing the simultaneity of the relation between the rate of inflation (πt) and the

nominal rate of interest (it). The coefficient of the lagged interest rate (it−1) is negative during the bubble period

of 1985-1990 where lower interest rate induce higher demand and higher rate of inflation. The coefficient of the

lagged rate of inflation (πt−1) represents the expected rate of inflation. Since 1997, the rate of inflation became

almost zero which shows high correlation with the lagged rate of inflation. The coefficient of lagged real GDP

(yt−1) shows positive during 1980 to 1987. On the other hand, after 1997, the ordinary Phillips Curve relations

can not be observed in Figure 3. (5) in Figure 3 denotes the reaction of the rate of inflation to the exchange rate.

When the exchange rate is appreciated, the export will decline and the rate of inflation will fall. Therefore the

expected sign ofet−1 is negative. Lastly the constant term (6) increased all the sudden in year 1997 where the

target values of the rate of inflation and the real growth rate had been dropped.

12

In Fig. 4, the results of Eq. (34) are shown. The coefficient ofit is positive since the real interest rate was rising

despite of the decline in nominal interest rate so that the real output did not rise much. From 2001, the growth rate

of real output became small so that the coefficient also shows almost zero. The coefficient ofπt−1 shows positive

since the real interest rate becomes lower and the real output rises. The coefficient ofyt−1 is negative from 1981

to 1995 since the real growth rate became lower and lower. Especially after 1998 the coefficient show almost

zero due to extremely low growth rate. The coefficient ofet−1 shows negative sign most of the period since the

appreciation of the yen lowered output growth.

In Fig. 5, the results of Eq. (36) are shown. In Fig. 5, the results of Eq. (35) are shown. The coefficients of

interest rates (namelyit andit−1), b4,1,0,t andb4,1,1,t, show positive sign in the entire period since appreciation of

the yen lowers interest rate. The coefficient of inflation rate (namelyπt andπt−1), b4,2,0,t andb4,2,1,t, show positive

and negative signs in the entire period. Since the rate of inflation,the real interest rate is lowered. Thus,output

increases and capital inflow from abroad. It makes Yen appreciated. The coefficient of inflation rate (namelyyt

andyt−1), b4,3,0,t andb4,3,1,t, show positive and negative signs in the entire period. Since the increases in output,

stock price is expected to rise. Thus, capital inflow increases and the Yen appreciated.

4 Conclusions and Discussions

In this paper, we present empirical analysis on Japanese monetary policy using Time-Varying Structural Vector

Autoregressions based on the Monte Carlo particle filter and a self-organizing state space model. We estimate

the time-varying reaction function of the Japanese monetary policy, and our TVSVAR also includes an aggregate

supply function, an aggregate demand function, and real exchange rate determination function. Our TVSVAR

is a dynamic full recursive structural VAR, which is similar to Primiceri (2005), Canova and Gambetti (2006),

and many related papers. The most of previous studies are based on Markov Chain Monte Carlo method and the

Kalman filter. Our approach, however, is based on the Monte Carlo Particle filter and a self-organizing state space

model, proposed by Kitagawa (1996), Gordon et al. (1993), Kitagawa (1998), Yano (2007b), and Yano (2007a).

The TVSVAR estimation method is proposed by Yano (2007c). In this paper, we assume on the time evolution

of coefficients which is depend on Markov chain. We call this assumptionMarkov chain priors. The Markov

chain priors are the generalization of random walk priors. Thus, the main feature of our method is less restrictions

rather than previous methods. Our methods are applied for the estimation of a quarterly model of the Japanese

economy (a nominal short term interest rate, the rate of inflation, the growth rate of real output, and the return of

the real effective exchange rate). We detect structural changes in most coefficients of TVSVAR. In effectiveness of

monetary policy by use of the interest rate can be seen in aggregate supply since 1990. Interest rate policy toward

aggregate demand is even worse in a sense that lower interest rate reduced output further. This paper concludes

the sluggish economy of Japan is caused not only by aggregate supply factor but also by aggregate demand factor.

Ineffectiveness of monetary policy since 1990 could not recover Japanese economy until recent.

For our future study, we would like to try thepth order of TVSVAR, estimating time-varying coefficients of

exogenous variables, and various type of TVSVAR. Moreover, we would like to try Time-Varying Structural VAR

with Stochastic Volatility (TVSVAR-SV) and Time-Varying Structural VAR with Sign Restrictions (TVSVAR-

SR). TVSVAR-SV is proposed by Primiceri (2005) and TVSVAR-SR is the dynamic version of Uhlig (2005)22.

22Braun and Shioji (2006) and Kamada and Sugo (2006) analyze the Japanese economy using sign-restricted VAR.

13

Appendix A Data Source

We use quarterly macroeconomic data of the Japanese economy from 1980:Q1 to 2006:Q3.

• Uncollateralized overnight call rate, averaged over three months (Bank Of Japan): uncollateralized overnight

call rate, monthly average (July 1985 - September 2006) and collateralized overnight call rate, monthly av-

erage (January 1980 - July 1985) are liked at July 1985.

http://www.boj.or.jp/en/theme/research/stat/market/index.htm

• Seasonal adjusted real/nominal GDP (Cabinet Office): Quarterly Estimates of GDP, chained, (1994:Q1-

2006:Q3) and Quarterly Estimates of GDP, fixed-based, (1980:Q1-1994:Q1) are linked at 1994:Q1.

http://www.esri.cao.go.jp/en/sna/menu.html

• Seasonal adjusted GDP deflator (Cabinet Office): deflator is calculated from seasonal adjusted real/nominal

GDP.

http://www.esri.cao.go.jp/en/sna/menu.html

• Real effective exchange rate (Bank Of Japan):

http://www.boj.or.jp/en/theme/research/stat/market/forex/index.htm

Appendix B Block Recursive TVSVAR

In subsection 3.1, the macroeconomic variables of FR-TVSVAR(1) are ordered from exogenous variables to en-

dogenous variables. This order of variables is a strong restriction on conventional SVAR. To release the restriction,

Christiano et al. (1999) propose “block-recursive” structural VAR. They partitionYt into three blocks:

Yt = [Xt, MP t, Zt]′,

whereXt is a non-monetary block,MPt is a monetary policy block,Zt is a monetary block.

We estimate the first order of block recursive TVSVAR(BR-TVSVAR(1)) as a benchmark model (from 1980:Q1

to 1998:Q4). BR-TVSVAR(1) is given by

yt = b1,1,1,tyt−1 + b1,2,1,tπt−1 + b1,3,1,tit−1 + b1,4,1,tet−1 + c1,t + ϵ1,t, (B1)

πt = b2,1,0,tyt + b2,1,1,tyt−1 + b2,2,1,tπt−1 + b2,3,1,tit−1 + b2,4,1,tet−1 + c2,t + ϵ2,t, (B2)

it = b3,1,0,tyt + b3,2,0,tπt + b3,1,1,tyt−1 + b3,2,1,tπt−1 + b3,3,1,tit−1 + b3,4,1,tet−1 + c3,t + ϵ3,t, (B3)

et = b4,1,0,tyt + b4,2,0,tπt + b4,3,0,tit + b4,1,1,tyt−1 + b4,2,1,tπt−1 + b4,3,1,tit−1 + b4,4,1,tet−1 + c4,t + ϵ4,t.

(B4)

whereit is the first difference of the short-term interest rate,πt is the rate of inflation,yt is the growth rate of real

output, andet is the return of the real effective exchange rate23.

23We set the number of particles,M , to 10000. The other parameters of simulation are same in Yano (2007b) and Yano

(2007a). All time-varying coefficients are standardized as follows:

bx,y,z,t = sdexp/sdobs,

wheresdexp is the standard deviation of an explaining variable andsdexp is the standard deviation of an observation (this

standardization method may not be best).

14





Appendix C Non-Gaussian State Space Model

We describe the detail of a non-Gaussian state space model. The time varying coefficientsbi,j,l,t anddi,n,t are

estimated by using MCPF. The non-Gaussian state space representation is given by

xt = Fxt−1 + Gvt,

yi,t = Htxt + ϵi,t,

i = 1, 2, · · · , k,

(C5)

whereF , G, Ht are(L×L), (L×L), and(1×L) matrices, respectively.xt is an(L× 1) vector of coefficients,

vt is anL variate possibly non-Gaussian noise,ϵi,t is a possibly non-Gaussian noise, andyi,t is an observation.

The symbolL is kp+n+ i−1. The detail of these vectors and matrices are explained in the following paragraphs.

In our algorithm, the matricesF , G are specified as follows.

F = IL, G = IL, (C6)

whereIL is anL-dimensional identity matrix.

For the convenience of the expression, we use the following notations:

bi,0,t =(bi,1,0,t, · · · , bi,i−1,0,t

),

bi,t =(bi,1,1,t, bi,2,1,t, · · · , bi,k,1,t,

bi,1,2,t, · · · , bi,k,2,t, · · · , bi,1,p,t, · · · , bi,k,p,t

),

di,t =(di,1,t, di,2,t, · · · , di,n,t

),

ri,t =(y1,t, y2,t, · · · , yi−1,t

),

ht =(y1,t, y2,t−1, · · · , yk,t−1, · · · ,

y1,t−p, y2,t−p, · · · , yk,t−p

),

ft =(u1,t−κ, u2,t−κ, · · · , un,t−κ

).

(C7)

The vectors,xt andHt are defined as follows. For the first component ofyt, i = 1,

xt =(b1,t, d1,t

)T,

Ht =(ht, ft

).

(C8)

For theith component ofy(t), 1 < i ≤ k,

xt =(bi,0,t, bi,t, di,t

)T,

Ht =(ri,t, ht, ft

).

(C9)

15

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18

List of Figures

1 Macroeconomics Data of Japan: 1980:I-2006:IV . . . . . . . . . . . . . . . . . . . . . 202 it = b1,1,1,tit−1 + b1,2,1,tπt−1 + b1,3,1,tyt−1 + b1,4,1,tet−1 + c1,t + ϵ1,t . . . . . . . . . . 213 πt = b2,1,0,tit + b2,1,1,tit−1 + b2,2,1,tπt−1 + b2,3,1,tyt−1 + b2,4,1,tet−1 + c2,t + ϵ2,t . . . . 224 yt = b3,1,0,tit +b3,2,0,tπt +b3,1,1,tit−1 +b3,2,1,tπt−1 +b3,3,1,tyt−1 +b3,4,1,tet−1 +c3,t + ϵ3,t 235 et = b4,1,0,tit+b4,2,0,tπt+b4,3,0,tyt+b4,1,1,tit−1+b4,2,1,tπt−1+b4,3,1,tyt−1+b4,4,1,tet−1+

c4,t + ϵ4,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 FR-TVSVAR(1): Impulse Response (1980:Q4) . . . . . . . . . . . . . . . . . . . . . . 257 FR-TVSVAR(1): Impulse Response (1985:Q4) . . . . . . . . . . . . . . . . . . . . . . 268 FR-TVSVAR(1): Impulse Response (1989:Q4) . . . . . . . . . . . . . . . . . . . . . . 279 FR-TVSVAR(1): Impulse Response (1997:Q4) . . . . . . . . . . . . . . . . . . . . . . 2810 Q-Q Plot (Full Recursive TVSVAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911 Autocorrelation with95% Confidence Interval (Full Recursive TVSVAR) . . . . . . . . 3012 it = b1,1,1,tit−1 + b1,2,1,tπt−1 + b1,3,1,tyt−1 + b1,4,1,tet−1 + c1,t + ϵ1,t . . . . . . . . . . 3113 πt = b2,1,0,tit + b2,1,1,tit−1 + b2,2,1,tπt−1 + b2,3,1,tyt−1 + b2,4,1,tet−1 + c2,t + ϵ2,t . . . . 3214 yt = b3,1,0,tit +b3,2,0,tπt +b3,1,1,tit−1 +b3,2,1,tπt−1 +b3,3,1,tyt−1 +b3,4,1,tet−1 +c3,t + ϵ3,t 3315 et = b4,1,0,tit+b4,2,0,tπt+b4,3,0,tyt+b4,1,1,tit−1+b4,2,1,tπt−1+b4,3,1,tyt−1+b4,4,1,tet−1+

c4,t + ϵ4,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416 Q-Q Plot (Full Recursive TVSVAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3517 Autocorrelation with95% Confidence Interval (Full Recursive TVSVAR) . . . . . . . . 3618 Q-Q Plot (Full Recursive SVAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3719 Autocorrelation with95% Confidence Interval (Full Recursive SVAR) . . . . . . . . . . 3820 Q-Q Plot (Full Recursive SVAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3921 Autocorrelation with95% Confidence Interval (Full Recursive SVAR) . . . . . . . . . . 4022 yt = b1,1,1,tyt−1 + b1,2,1,tπt−1 + b1,3,1,tit−1 + b1,4,1,tet−1 + c1,t + ϵ1,t . . . . . . . . . . 4123 πt = b2,1,0,tyt + b2,1,1,tyt−1 + b2,2,1,tπt−1 + b2,3,1,tit−1 + b2,4,1,tet−1 + c2,t + ϵ2,t . . . 4224 it = b3,1,0,tyt +b3,2,0,tπt +b3,1,1,tyt−1 +b3,2,1,tπt−1 +b3,3,1,tit−1 +b3,4,1,tet−1 +c3,t + ϵ3,t 4325 et = b4,1,0,tyt+b4,2,0,tπt+b4,3,0,tit+b4,1,1,tyt−1+b4,2,1,tπt−1+b4,3,1,tit−1+b4,4,1,tet−1+

c4,t + ϵ4,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

19

−2−1

01

2

Cal

l Rat

e

−10

12

Infla

tion

Rat

e

−2−1

01

2

Rea

l Gro

wth

−10

05

10

1980 1985 1990 1995 2000 2005

Exch

ange

Rat

e

Time

Figure 1: Macroeconomics Data of Japan: 1980:I-2006:IV

20

(1) b1,1,1,t (it−1)

Time

1980 1985 1990 1995

0.0

0.4

0.8

(2) b1,2,1,t (πt−1)

Time

1980 1985 1990 1995

0.2

0.4

0.6

(3) b1,3,1,t (yt−1)

Time

1980 1985 1990 1995

−0

.6−

0.2

(4) b1,4,1,t (et−1)

Time

1980 1985 1990 1995−

3−

10

(5) c1,t

Time

1980 1985 1990 1995

−0

.3−

0.1

0.1

Figure 2:it = b1,1,1,tit−1 + b1,2,1,tπt−1 + b1,3,1,tyt−1 + b1,4,1,tet−1 + c1,t + ϵ1,t

21

(1) b2,1,0,t (it)

Time

1980 1985 1990 1995

0.0

0.4

(2) b2,1,1,t (it−1)

Time

1980 1985 1990 1995

−0

.8−

0.2

(3) b2,2,1,t (πt−1)

Time

1980 1985 1990 1995

−0

.5−

0.1

0.2

(4) b2,3,1,t (yt−1)

Time

1980 1985 1990 1995

−0

.20

.2

(5)b2,4,1,t (et−1)

Time

1980 1985 1990 1995

−3

−1

1

(6)c2,t

Time

1980 1985 1990 1995

0.0

0.4

Figure 3:πt = b2,1,0,tit + b2,1,1,tit−1 + b2,2,1,tπt−1 + b2,3,1,tyt−1 + b2,4,1,tet−1 + c2,t + ϵ2,t

22

(1) b3,1,0,t (it)

Time

1980 1985 1990 1995

−0.6

−0.2

0.2

(2) b3,2,0,t (πt)

Time

1980 1985 1990 1995

−0.2

0.2

0.6

(3) b3,1,1,t (it−1)

Time

1980 1985 1990 1995

−0.5

0.0

0.5

1.0

(4) b3,2,1,t (πt−1)

Time

1980 1985 1990 1995−0

.8−0

.40.

0

(5)b3,3,1,t (yt−1)

Time

1980 1985 1990 1995

−0.8

−0.4

0.0

(6)b3,4,1,t (et−1)

Time

1980 1985 1990 1995

−20

24

(7)c3,t

Time

1980 1985 1990 1995

−0.5

0.5

1.5

Figure 4:yt = b3,1,0,tit + b3,2,0,tπt + b3,1,1,tit−1 + b3,2,1,tπt−1 + b3,3,1,tyt−1 + b3,4,1,tet−1 + c3,t + ϵ3,t

23

(1) b4,1,0,t (it)

Time

1980 1985 1990 1995

−0.6

−0.2

0.2

(2) b4,2,0,t (πt)

Time

1980 1985 1990 1995

−0.6

−0.2

0.2

(3) b4,3,0,t (yt)

Time

1980 1985 1990 1995

−0.8

−0.4

0.0

0.4

(4) b4,1,1,t (it−1)

Time

1980 1985 1990 19950.

00.

20.

40.

6

(5)b4,2,1,t (πt−1)

Time

1980 1985 1990 1995

−0.6

−0.2

0.2

(6)b4,3,1,t (yt−1)

Time

1980 1985 1990 1995

−1.0

−0.5

0.0

0.5

(7)b4,4,1,t (et−1)

Time

1980 1985 1990 1995

−1.0

0.0

1.0

2.0

(8)c4,t

Time

1980 1985 1990 1995

−10

12

3

Figure 5:et = b4,1,0,tit + b4,2,0,tπt + b4,3,0,tyt + b4,1,1,tit−1 + b4,2,1,tπt−1 + b4,3,1,tyt−1 + b4,4,1,tet−1 +c4,t + ϵ4,t

24

2 6 10

0.0

1.0

i

Quarter

i

2 6 10

−0.

5

i −−>π

Quarter

2 6 10

−0.

30

i −−> y

Quarter

2 6 10

−0.

8

i −−> e

Quarter

2 6 10

0.0

0.6

π −−> i

Quarter

π

2 6 10

0.0

1.0

π

Quarter

2 6 10

0.00

π −−> y

Quarter

2 6 10

−1.

2

π −−> e

Quarter

2 6 10

−0.

3

y −−> i

Quarter

y

2 6 10

0.00

y −−>π

Quarter

2 6 10

0.0

1.0

y

Quarter

2 6 10

−0.

10

y −−> e

Quarter

2 6 10

−0.

10

e −−> i

Quarter

e

2 6 10

−0.

04

e −−>π

Quarter

2 6 10

0.00

e −−> y

Quarter

2 6 10

0.0

1.0

e

Quarter

Figure 6: FR-TVSVAR(1): Impulse Response (1980:Q4)

25

2 6 10

0.0

1.0

i

Quarter

i

2 6 10

−0.

25

i −−>π

Quarter

2 6 10

−0.

3

i −−> y

Quarter

2 6 10

0.0

i −−> e

Quarter

2 6 10

0.00

π −−> i

Quarter

π

2 6 10

0.0

1.0

π

Quarter

2 6 10

0.0

π −−> y

Quarter

2 6 10

0.0

0.8

π −−> e

Quarter

2 6 10

−0.

20

y −−> i

Quarter

y

2 6 10

0.00

y −−>π

Quarter

2 6 10

−0.

5y

Quarter

2 6 10

−0.

60.

0

y −−> e

Quarter

2 6 10

−0.

020

e −−> i

Quarter

e

2 6 10

0.00

e −−>π

Quarter

2 6 10

0.00

e −−> y

Quarter

2 6 10

0.0

1.0

e

Quarter


26

2 6 10

0.0

1.0

i

Quarter

i

2 6 10

0.00

i −−>π

Quarter

2 6 10

0.00

i −−> y

Quarter

2 6 10

02

i −−> e

Quarter

2 6 10

0.00

π −−> i

Quarter

π

2 6 10

−0.

41.

0π

Quarter

2 6 10

0.0

0.5

π −−> y

Quarter

2 6 10

0.0

1.5

π −−> e

Quarter

2 6 10

−0.

035

y −−> i

Quarter

y

2 6 10

0.00

y −−>π

Quarter

2 6 10

−0.

41.

0y

Quarter

2 6 10

0.0

1.0

y −−> e

Quarter

2 6 10

0.00

0

e −−> i

Quarter

e

2 6 10

−0.

03

e −−>π

Quarter

2 6 10

−0.

06

e −−> y

Quarter

2 6 10

0.0

1.0

e

Quarter


27

2 6 10

0.0

1.0

i

Quarter

i

2 6 10

−0.

10

i −−>π

Quarter

2 6 10

0.0

1.2

i −−> y

Quarter

2 6 10

02

i −−> e

Quarter

2 6 10

0.00

π −−> i

Quarter

π

2 6 10

−0.

5π

Quarter

2 6 10

0.0

0.8

π −−> y

Quarter

2 6 10

0.0

3.0

π −−> e

Quarter

2 6 10

0.00

y −−> i

Quarter

y

2 6 10

0.00

y −−>π

Quarter

2 6 10

−0.

5y

Quarter

2 6 10

−1.

5

y −−> e

Quarter

2 6 10

−0.

014

e −−> i

Quarter

e

2 6 10

−0.

15

e −−>π

Quarter

2 6 10

0.00

e −−> y

Quarter

2 6 10

0.0

1.0

e

Quarter


28

−2 −1 0 1 2

−0

.06

−0

.02

0.0

2

εi

Theoretical Quantiles

Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.20

.00

.2

επ


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.10

.00

.10

.2

εy


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−1

.0−

0.5

0.0

0.5

εe


Sa

mp

le Q

ua

ntil

es

Figure 10: Q-Q Plot (Full Recursive TVSVAR)

29

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεi

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

επ

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

εy

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεe

Figure 11: Autocorrelation with95% Confidence Interval (Full Recursive TVSVAR)

30

(1) b1,1,1,t (it−1)

Time

1980 1985 1990 1995 2000 2005

0.0

0.4

0.8

(2) b1,2,1,t (πt−1)

Time

1980 1985 1990 1995 2000 2005

0.2

0.5

0.8

(3) b1,3,1,t (yt−1)

Time

1980 1985 1990 1995 2000 2005

−0

.8−

0.4

0.0

(4) b1,4,1,t (et−1)

Time

1980 1985 1990 1995 2000 2005−

3−

1

(5) c1,t

Time

1980 1985 1990 1995 2000 2005

−0

.3−

0.1

0.1

Figure 12:it = b1,1,1,tit−1 + b1,2,1,tπt−1 + b1,3,1,tyt−1 + b1,4,1,tet−1 + c1,t + ϵ1,t

31

(1) b2,1,0,t (it)

Time

1980 1985 1990 1995 2000 2005

−0

.10

.20

.5

(2) b2,1,1,t (it−1)

Time

1980 1985 1990 1995 2000 2005

−0

.6−

0.2

(3) b2,2,1,t (πt−1)

Time

1980 1985 1990 1995 2000 2005

−0

.4−

0.1

0.2

(4) b2,3,1,t (yt−1)

Time

1980 1985 1990 1995 2000 2005

−0

.40

.00

.4

(5)b2,4,1,t (et−1)

Time

1980 1985 1990 1995 2000 2005

−3

−1

01

(6)c2,t

Time

1980 1985 1990 1995 2000 2005

−0

.60

.00

.4

Figure 13:πt = b2,1,0,tit + b2,1,1,tit−1 + b2,2,1,tπt−1 + b2,3,1,tyt−1 + b2,4,1,tet−1 + c2,t + ϵ2,t

32

(1) b3,1,0,t (it)

Time

1980 1990 2000

−0.5

0.0

0.5

1.0

(2) b3,2,0,t (πt)

Time

1980 1990 2000

−1.0

0.0

0.5

1.0

(3) b3,1,1,t (it−1)

Time

1980 1990 2000

−0.5

0.0

0.5

1.0

(4) b3,2,1,t (πt−1)

Time

1980 1990 2000−0

.8−0

.40.

00.

4

(5)b3,3,1,t (yt−1)

Time

1980 1990 2000

−1.0

−0.5

0.0

0.5

(6)b3,4,1,t (et−1)

Time

1980 1990 2000

−4−2

02

4

(7)c3,t

Time

1980 1990 2000

−0.5

0.5

1.5

Figure 14:yt = b3,1,0,tit + b3,2,0,tπt + b3,1,1,tit−1 + b3,2,1,tπt−1 + b3,3,1,tyt−1 + b3,4,1,tet−1 + c3,t + ϵ3,t

33

(1) b4,1,0,t (it)

Time

1980 1990 2000

−0.6

−0.2

(2) b4,2,0,t (πt)

Time

1980 1990 2000

−0.5

0.0

0.5

(3) b4,3,0,t (yt)

Time

1980 1990 2000

−1.0

−0.5

0.0

0.5

(4) b4,1,1,t (it−1)

Time

1980 1990 20000.

00.

20.

40.

6

(5)b4,2,1,t (πt−1)

Time

1980 1990 2000

−0.5

0.0

0.5

(6)b4,3,1,t (yt−1)

Time

1980 1990 2000

−1.0

0.0

0.5

1.0

(7)b4,4,1,t (et−1)

Time

1980 1990 2000

−2−1

01

2

(8)c4,t

Time

1980 1990 2000

−20

12

34

5

Figure 15:et = b4,1,0,tit + b4,2,0,tπt + b4,3,0,tyt + b4,1,1,tit−1 + b4,2,1,tπt−1 + b4,3,1,tyt−1 + b4,4,1,tet−1 +c4,t + ϵ4,t

34

−2 −1 0 1 2

−0

.3−

0.1

0.0

0.1

εi


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.20

.00

.2

επ


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.3−

0.1

0.1

0.3

εy


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.02

0.0

20

.06

εe


Sa

mp

le Q

ua

ntil

es

Figure 16: Q-Q Plot (Full Recursive TVSVAR)

35

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεi

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

επ

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

εy

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεe

Figure 17: Autocorrelation with95% Confidence Interval (Full Recursive TVSVAR)

36

−2 −1 0 1 2

−1

.00

.00

.51

.0

εi


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.50

.51

.0

επ


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−2

−1

01

εy


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−1

00

51

0

εe


Sa

mp

le Q

ua

ntil

es

Figure 18: Q-Q Plot (Full Recursive SVAR)

37

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεi

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

επ

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

εy

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεe

Figure 19: Autocorrelation with95% Confidence Interval (Full Recursive SVAR)

38

−2 −1 0 1 2

−1

.00

.01

.0

εi


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−0

.50

.51

.5

επ


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−2

−1

01

εy


Sa

mp

le Q

ua

ntil

es

−2 −1 0 1 2

−1

00

51

0

εe


Sa

mp

le Q

ua

ntil

es

Figure 20: Q-Q Plot (Full Recursive SVAR)

39

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεi

0 2 4 6 8

−0

.40

.00

.40

.8

Lag

AC

F

επ

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

F

εy

0 2 4 6 8

−0

.20

.20

.61

.0

Lag

AC

Fεe

Figure 21: Autocorrelation with95% Confidence Interval (Full Recursive SVAR)

40

(1) b1,1,1,t (yt−1)

Time

1980 1985 1990 1995

−0

.30

.00

.2

(2) b1,2,1,t (πt−1)

Time

1980 1985 1990 1995

−0

.20

.00

.2

(3) b1,3,1,t (it−1)

Time

1980 1985 1990 1995

−0

.20

.10

.3

(4) b1,4,1,t (et−1)

Time

1980 1985 1990 1995−

0.0

50

.05

(5) c1,t

Time

1980 1985 1990 1995

0.0

0.5

1.0

1.5

Figure 22:yt = b1,1,1,tyt−1 + b1,2,1,tπt−1 + b1,3,1,tit−1 + b1,4,1,tet−1 + c1,t + ϵ1,t

41

(1) b2,1,0,t (yt)

Time

1980 1985 1990 1995

0.0

0.4

0.8

(2) b2,1,1,t (yt−1)

Time

1980 1985 1990 1995

0.0

0.4

(3) b2,2,1,t (πt−1)

Time

1980 1985 1990 1995

−0

.6−

0.2

0.2

(4) b2,3,1,t (it−1)

Time

1980 1985 1990 1995

−0

.7−

0.4

−0

.1

(5)b2,4,1,t (et−1)

Time

1980 1985 1990 1995

−3

−1

1

(6)c2,t

Time

1980 1985 1990 1995

−0

.05

0.0

5

Figure 23:πt = b2,1,0,tyt + b2,1,1,tyt−1 + b2,2,1,tπt−1 + b2,3,1,tit−1 + b2,4,1,tet−1 + c2,t + ϵ2,t

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(1) b3,1,0,t (yt)

Time

1980 1985 1990 1995

−1.0

0.0

0.5

(2) b3,2,0,t (πt)

Time

1980 1985 1990 1995

0.2

0.6

1.0

(3) b3,1,1,t (yt−1)

Time

1980 1985 1990 1995

−1.0

−0.5

0.0

0.5

(4) b3,2,1,t (πt−1)

Time

1980 1985 1990 1995−0

.40.

00.

40.

8

(5)b3,3,1,t (it−1)

Time

1980 1985 1990 1995

−0.2

0.2

0.6

(6)b3,4,1,t (et−1)

Time

1980 1985 1990 1995

−4−2

01

(7)c3,t

Time

1980 1985 1990 1995

−0.8

−0.4

0.0

Figure 24:it = b3,1,0,tyt + b3,2,0,tπt + b3,1,1,tyt−1 + b3,2,1,tπt−1 + b3,3,1,tit−1 + b3,4,1,tet−1 + c3,t + ϵ3,t

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(1) b4,1,0,t (yt)

Time

1980 1985 1990 1995

−0.5

0.0

0.5

(2) b4,2,0,t (πt)

Time

1980 1985 1990 1995

−0.8

−0.4

0.0

(3) b4,3,0,t (it)

Time

1980 1985 1990 1995

−0.8

−0.4

0.0

(4) b4,1,1,t (yt−1)

Time

1980 1985 1990 1995−1

.0−0

.6−0

.20.

2

(5)b4,2,1,t (πt−1)

Time

1980 1985 1990 1995

−1.0

−0.5

0.0

0.5

(6)b4,3,1,t (it−1)

Time

1980 1985 1990 1995

0.0

0.4

0.8

(7)b4,4,1,t (et−1)

Time

1980 1985 1990 1995

−2−1

01

2

(8)c4,t

Time

1980 1985 1990 1995

−20

24

6

Figure 25:et = b4,1,0,tyt + b4,2,0,tπt + b4,3,0,tit + b4,1,1,tyt−1 + b4,2,1,tπt−1 + b4,3,1,tit−1 + b4,4,1,tet−1 +c4,t + ϵ4,t

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List of Tables

1 Root Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Root Mean Square Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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Table 1: Root Mean Square Error

RMSE Time-Varying SVAR Fixed Coefficient SVARϵi 0.01950 0.44802ϵπ 0.10686 0.50403ϵy 0.04861 0.89696ϵe 0.24998 4.15722

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Table 2: Root Mean Square Error

RMSE Time-Varying SVAR Fixed Coefficient SVARϵi 0.04303 0.45066ϵπ 0.11014 0.52432ϵy 0.08479 0.89801ϵe 0.08479 4.15124

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Japanese Monetary Policy Reaction Function and Time-Varying … · 2017. 3. 6. · Japanese...

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Transcript of Japanese Monetary Policy Reaction Function and Time-Varying … · 2017. 3. 6. · Japanese...